Polarization algebras and their relations

Abstract

Using an approach to the Jacobian conjecture by L.M. Druzkowski and K. Rusek, G. Gorni and G. Zampieri, and A.V. Yagzhev, we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert's problem in classical ring theory and the homogeneous dependence problem in affine algebraic geometry related to the Jacobian conjecture. We demonstrate these relations in concrete examples and formulate some open questions.